Independent Set Hardness in Graphs of Bounded Twin-Width and Low-Radius Merge-Width

For every $\varepsilon > 0$, Max Independent Set admits a polynomial-time $n^\varepsilon$-approximation algorithm on $n$-vertex graphs of effectively bounded twin-width [Berg\'e et al., STACS '23]. The approximation factor actually obtained is more precisely $n^{O(1/ \log \log n)}$. Prior to the current paper, no approximation hardness was known for this problem, and the existence of a polynomial-time approximation scheme (PTAS) was repeatedly raised as an open question. We answer this question in a strong sense: We show that there is a constant $\gamma > 0$ such that a polynomial-time $n^{\gamma/ (\log \log n)^2}$-approximation algorithm for Max Independent Set on graphs of twin-width at most 4 would refute the Exponential-Time Hypothesis (ETH). This lower bound further holds if a 4-sequence is provided as part of the input. We show the same hardness of approximation for Min Coloring, which also has a nearly matching $n^{O(1/ \log \log n)}$-approximation algorithm on graphs of effectively bounded twin-width. We also clarify the parameterized complexity of $k$-Independent Set on graphs of bounded radius-$r$ merge-width when the range of $r$ is limited. There is a fixed-parameter tractable algorithm for $k$-Independent Set on graphs given with radius-$2^{O(k^2)}$ merge sequences of bounded width [Dreier and Toru\'nczyk, STOC '25]. We complement this result by showing that $k$-Independent Set is W[1]-hard on graphs given with radius-$o(k)$ merge sequences of bounded width. We further show that this result also holds for $k$-Dominating Set.